What is the ruling on someone who felt something exit that would invalidate their prayer, and then found a drop of water after finishing their prayer; and must they repeat their prayer?

If you are not certain that something has exited, and what is found on the clothes could possibly be from the traces of istinja' (cleaning oneself after relieving oneself), then do not pay attention to the doubt, and build upon the original principle, which is that nothing has exited. However, if you are absolutely certain that something exited during the prayer, then you must repeat it. It is good to sprinkle water on the clothes after istinja' to ward off intrusive thoughts (waswasa).